Combining Cuisenaire

4515 /www/nrich/html/content/id/4515/ 1 2011-02-01T00:00:01 <?xml version="1.0" encoding="UTF-8"?> <mdoxml version="1.0"> Here are two different Cuisenaire rods: <mdo:image width="201" height="21" src="tworods.gif" alt="two different rods"></mdo:image> How many different ways can you find to line them up end to end? How many different ways are there to line them up if both rods are the same? Now there are three different rods: <mdo:image width="221" height="21" alt="three rods" src="threerods.gif"></mdo:image> In how many different ways can you line up the three rods? Imagine two of the rods are the same. How many different ways are there to line them up now? How many different ways are there if all the rods are the same? What would happen if there were four different rods? How many ways can you line them up now? If two rods are the same, how many ways could you line them up? And how many ways are there to line them up if three are the same? How about if they are all the same? You might like to use the interactivity below to try out your ideas. <a href="/content/id/4515/cuisenaire.swf">Full Screen Version</a> <mdo:flash height="400" width="550"><param name="movie" value="/content/id/4515/cuisenaire.swf" ></param><param name="flashplayerversion" value="7" ></param><param name="height" value="400" ></param><param name="width" value="550" ></param></mdo:flash> </mdoxml> <?xml version="1.0" encoding="UTF-8"?> <mdoxml version="1.0"> Emma from St Paul's Girls' School sent in a very good solution. She says: For two differently coloured rods, there are 2 ways you can line them up end to end: R-Y Y-R If the two rods are the same colour however there is only 1 way in which they can be lined up end to end: R-R (or Y-Y depending on what colour they are) For three differently coloured rods, there are 6 ways to line them up end to end: R-G-P R-P-G P-G-R P-R-G G-P-R G-R-P If two of these rods are the same colour - let them be red - there are 3 ways in which the rods can be lined up end to end: R-R-P R-P-R P-R-R If all the rods are the same colour - let them be red - there is only 1 way in which they can be lined up end to end: R-R-R When there are four differently coloured rods there are 24 ways in which they can be lined up end to end: A-B-C-D A-B-D-C A-C-B-D A-C-D-B A-D-B-C A-D-C-B x4 (because of when B, C and D are first in the line of rods) When two of the rods are the same colour there are 18 ways in which the four rods can be lined end to end: A-A-B-C A-A-C-B A-B-A-C A-B-C-A A-C-A-B A-C-B-A x3 (because of when B and C are first in the line) Are you sure that when B is first in the line there will be six ways, Emma? Remember that there are two As... Perhaps someone else can help here? When three of the four rods are the same colour there are 4 ways they can be placed end to end: A-A-A-B A-A-B-A A-B-A-A B-A-A-A <div style="clear: both;">When they are all the same colour there is again only one way in which the rods can be lined: <div style="clear: both;">A-A-A-A</div><div style="clear: both;"></div> Very well done, Emma, you have gone about this in a very ordered way. </div> </mdoxml> <?xml version="1.0" encoding="UTF-8"?> <mdoxml version="1.0"><div class="embed"> <h2>Combining Cuisenaire</h2> Here are two different Cuisenaire rods: <mdo:image alt="two different rods" height="21" src="tworods.gif" width="201"></mdo:image> How many different ways can you find to line them up end to end? How many different ways are there to line them up if both rods are the same? Now there are three different rods: <mdo:image alt="three rods" height="21" src="threerods.gif" width="221"></mdo:image> In how many different ways can you line up the three rods? Imagine two of the rods are the same. How many different ways are there to line them up now? How many different ways are there if all the rods are the same? What would happen if there were four different rods? How many ways can you line them up now? If two rods are the same, how many ways could you line them up? And how many ways are there to line them up if three are the same? How about if they are all the same? You might like to use the interactivity below to try out your ideas. <a href="/content/id/4515/cuisenaire.swf">Full Screen Version</a> <mdo:flash height="400" id="/content/id/4515/cuisenaire.swf" width="550"><param name="allowfullscreen" value="true"></param><param name="movie" value="/content/id/4515/cuisenaire.swf"></param> <param name="flashplayerversion" value="7"></param> <param name="height" value="400"></param> <param name="width" value="550"></param> </mdo:flash></div> This problem is a very basic introduction to combinations and permutations. The hints would make good questions to ask children once they have had a first go and encouraging them to talk about what they are doing is invaluable. Using "real" Cuisenaire rods (and OHT rods for demonstration purposes) will give children a chance to try out ideas if they are not able to use the interactivity. One of the main aims of this problem is to give pupils the opportunity to work in a systematic way, which might need discussion and modelling. </mdoxml> <?xml version="1.0" encoding="UTF-8"?> <mdoxml version="1.0">How will you know that you have all the different ways? If you're not using the interactivity, how will you record the ways you find? </mdoxml> <?xml version="1.0" encoding="UTF-8"?> <mdoxml version="1.0"> 2 different rods: 2 2 the same: 1 3 different rods: 6 3 rods with 2 the same: 3 3 the same: 1 4 different rods: 24 4 rods with 2 the same: 12 4 rods with 3 the same:4 4 rods the same: 1 </mdoxml> 2 4 0 1 0 0 0 Combining Cuisenaire Can you find all the different ways of lining up these Cuisenaire rods? Using, Applying and Reasoning about Mathematics Working systematically Decision Mathematics and Combinatorics Combinations Mathematics Tools Cuisenaire rods Information and Communications Technology Interactivities